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In this paper, we consider the numerical solution of the time fractional diffusion equation. Essentially, the time fractional diffusion equation differs from the standard diffusion equation in the time derivative term. In the former case, the first-order time derivative is replaced by a fractional derivative, making the problem global in time. We propose a spectral method in both temporal and spatial discretizations for this equation. The convergence of the method is proven by providing a priori error estimate. Numerical tests are carried out to confirm the theoretical results. Thanks to the spectral accuracy in both space and time of the proposed method, the storage requirement due to the "global time dependence" can be considerably relaxed, and therefore calculation of the long-time solution becomes possible.

Numerical methods for solving the continuum model of the dynamics of the molecular beam epitaxy (MBE) require very large time simulation, and therefore large time steps become necessary. The main purpose of this work is to construct and analyze highly stable time discretizations which allow much larger time steps than those of a standard implicit-explicit approach. To this end, an extra term, which is consistent with the order of the time discretization, is added to stabilize the numerical schemes. Then the stability properties of the resulting schemes are established rigorously. Numerical experiments are carried out to support the theoretical claims. The proposed methods are also applied to simulate the MBE models with large solution times. The power laws for the coarsening process are obtained and are compared with previously published results. Introduction.Recently there has been significant research interest in the dynamics of molecular beam epitaxy (MBE) growth. The MBE technique is among the most refined methods for the growth of thin solid films, and it is of great importance for applied studies; see, e.g., [1,16,22]. The evolution of the surface morphology during epitaxial growth results in a delicate relation between the molecular flux and the relaxation of the surface profile through surface diffusion of adatoms. It occurs on time and length scales that may span several orders of magnitude. Different kinds of models have been used to describe such phenomena; these typically include atomistic models, continuum models, and hybrid models. The atomistic models are usually implemented in the form of molecular dynamics or kinetic Monte Carlo simulations [4,9,17]. The continuum models are based on partial differential equations and are appropriate mainly for investigating the temporal evolution of the MBE instability at large time and length scales [11,24]. The hybrid models can be considered as a compromise between atomistic models and continuum models; see, e.g., [3,8].We are interested in the continuum models for the evolution of the MBE growth. Let h(x, t) be the epitaxy surface height with x ∈ R 2 and t ≥ 0. Under typical conditions for MBE growth, the height evolution equation can be written under mass conservation form (see, e.g., [14]):

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Abstract. The Cable equation has been one of the most fundamental equations for modeling neuronal dynamics. In this paper, we consider the numerical solution of the fractional Cable equation, which is a generalization of the classical Cable equation by taking into account the anomalous diffusion in the movement of the ions in neuronal system. A schema combining a finite difference approach in the time direction and a spectral method in the space direction is proposed and analyzed. The main contribution of this work is threefold: 1) We construct a finite difference/Legendre spectral schema for discretization of the fractional Cable equation. 2) We give a detailed analysis of the proposed schema by providing some stability and error estimates. Based on this analysis, the convergence of the method is rigourously established. We prove that the overall schema is unconditionally stable, and the numerical solution converges to the exact one with order O( t 2−max{α,β} + t −1 N −m ), where t, N and m are respectively the time step size, polynomial degree, and regularity in the space variable of the exact solution. α and β are two different exponents between 0 and 1 involved in the fractional derivatives. 3) Finally, some numerical experiments are carried out to support the theoretical claims.

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